Math Problem Statement
Certainly! Here’s the translation:
"2. Given that one vertex of a rectangle is at (2,1), another vertex is at (6,5), and the length of the diagonal is 4 units, find the coordinates of the remaining two vertices of the rectangle."
Solution
To solve this problem, we start by understanding the given information:
- Two vertices of the rectangle are at points and .
- The diagonal length of the rectangle is 4 units.
- We need to find the coordinates of the remaining two vertices, say and .
Step-by-Step Solution
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Diagonal Length Calculation: The length of the diagonal of a rectangle can be found using the distance formula. Given that and are two opposite vertices of the rectangle, let's calculate the diagonal length:
Since the length of diagonal is , but it is given that the diagonal length is 4 units, cannot be the diagonal. Thus, and must be adjacent vertices.
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Finding Midpoint and Using Diagonal Properties: For the two unknown vertices and , let's denote:
Since the diagonals of a rectangle bisect each other, the midpoint of diagonal is also the midpoint of diagonal . Let's find the midpoint of :
This means that the midpoint of is also .
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Using Distance to Find Remaining Vertices: We know that the diagonal length . Let’s use the distance formula to express the coordinates of points and in terms of their distance to the midpoint .
Squaring both sides:
These equations represent two circles of radius 2 centered at .
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Determine Specific Coordinates: Since and are symmetric about point , we can express the solutions as:
Substituting into the distance formula gives:
Choosing and , we find:
- For , the points are and .
- For , the points are and .
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Conclusion: The coordinates of the remaining vertices are and or and .
Would you like more details or have any questions?
Related Questions:
- How do you find the midpoint of a line segment?
- What are the properties of diagonals in a rectangle?
- How do you determine if two points are adjacent or opposite vertices of a rectangle?
- Can you explain the distance formula and its applications in geometry?
- How do you solve for unknown variables using geometric constraints?
Tip:
When working with geometric shapes, visualizing the problem by sketching can often make it easier to understand and solve.
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Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Rectangles
Formulas
Distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Theorems
Properties of diagonals in a rectangle
Pythagorean Theorem
Suitable Grade Level
Grades 9-11